answer only Find all the zeros of
𝑃(𝑥)=2𝑥5+3𝑥4−44𝑥3−23𝑥2+102𝑥−40

To find the zeros of the polynomial \( P(x) = 2x^5 + 3x^4 - 44x^3 - 23x^2 + 102x - 40 \), we can use the Rational Root Theorem, synthetic division, and factoring techniques.

Let's follow these steps to find the zeros:

1. **Use the Rational Root Theorem**: The possible rational roots are factors of the constant term \( -40 \) divided by factors of the leading coefficient \( 2 \). These are \( \pm 1, \pm 2, \pm 4, \pm 5, \pm 8, \pm 10, \pm 20, \pm 40 \) divided by \( \pm 1, \pm 2 \).

So, the possible rational roots are \( \pm 1, \pm 2, \pm 4, \pm 5, \pm 8, \pm 10, \pm 20, \pm 40, \pm \frac{1}{2}, \pm \frac{5}{2}, \pm \frac{10}{2}, \pm \frac{20}{2}, \pm \frac{40}{2} \).

2. **Test for roots** using synthetic division or direct substitution.

- By testing \( x = 2 \), we find it is a root.

Thus, \( P(x) \) is divisible by \( (x - 2) \).

3. **Perform synthetic division** of \( P(x) \) by \( (x - 2) \) to simplify the polynomial.

After performing synthetic division, the quotient is:
\[ 2x^4 + 7x^3 - 30x^2 - 83x + 20 \]

4. **Repeat the process** to find other roots:

By testing more values (or continuing synthetic division), we find another root: \( x = -4 \).

Dividing \( 2x^4 + 7x^3 - 30x^2 - 83x + 20 \) by \( (x + 4) \), we get:
\[ 2x^3 - x^2 - 14x - 5 \]

5. **Solve the cubic equation** using factoring or other techniques. The cubic equation \( 2x^3 - x^2 - 14x - 5 \) can be solved, and its roots are \( x = 5, x = -\frac{1}{2}, x = -1 \).

### Final Answer:
The zeros of \( P(x) = 2x^5 + 3x^4 - 44x^3 - 23x^2 + 102x - 40 \) are:
\[
x = 2, -4, 5, -\frac{1}{2}, -1
\]

Would you like further details on the steps?

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1. What is the Rational Root Theorem, and how does it apply to this problem?
2. How is synthetic division used in polynomial division?
3. What are the possible rational roots of a polynomial, and how are they determined?
4. How can we solve a cubic equation after factoring?
5. What are the benefits of factoring a polynomial to find its zeros?

**Tip**: Synthetic division is a faster alternative to long division when testing for polynomial roots.

Find all the zeros of P(x) = 2x^5 + 3x^4 - 44x^3 - 23x^2 + 102x - 40

Learn how to find all the zeros of the polynomial P(x) = 2x^5 + 3x^4 - 44x^3 - 23x^2 + 102x - 40 using the Rational Root Theorem, synthetic division, and factoring techniques. This problem covers key algebraic concepts such as polynomial functions, factoring, and solving cubic equations, suitable for Grades 10-12.

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